Arithmetic sequences
An arithmetic sequence in Grade 7 is a number pattern where the same value is added to each term to get the next, resulting in numbers equally spaced on a number line. In Saxon Math, Course 2, examples include the odd numbers 1, 3, 5, 7, ... (add 2 each time) and the decreasing sequence 20, 15, 10, 5, ... (add -5 each time). The constant difference between consecutive terms is called the common difference. Recognizing and extending arithmetic sequences builds the foundation for linear functions, slope, and algebraic equations.
Key Concepts
Property In an arithmetic sequence , the same number is added to each term to find the next term. The numbers in the sequence are equally spaced on a number line.
Examples The sequence of odd numbers $1, 3, 5, 7, ...$ is arithmetic because you consistently add 2. The sequence $20, 15, 10, 5, ...$ is arithmetic because the rule is to add $ 5$ (or subtract 5) each time. To find the next term of $100, 110, 120, ...$, you see the rule is 'add 10', so the next term is $120 + 10 = 130$.
Explanation Imagine you're climbing a staircase where every single step is the exact same height. That's an arithmetic sequence! You just keep adding the same number over and over again, making it a steady and predictable climb. This constant step, called the common difference, is the secret to finding any term in the sequence, no matter how far along.
Common Questions
What is an arithmetic sequence?
An arithmetic sequence is an ordered list where each term is obtained by adding the same constant (the common difference) to the previous term. The terms are equally spaced on a number line.
How do you find the common difference of an arithmetic sequence?
Subtract any term from the next term. If the result is the same throughout, that value is the common difference. For example, in 5, 8, 11, 14, the common difference is 8 - 5 = 3.
Can an arithmetic sequence decrease?
Yes. If the common difference is negative (you subtract the same amount each time), the sequence decreases. For example, 20, 15, 10, 5, ... has a common difference of -5.
How do you find the next term in an arithmetic sequence?
Add the common difference to the last known term. For example, in 1, 3, 5, 7, ..., the common difference is 2, so the next term is 7 + 2 = 9.
Where are arithmetic sequences taught in Saxon Math Course 2?
Arithmetic sequences are introduced in Saxon Math, Course 2, as part of Grade 7 patterns, functions, and algebraic reasoning.
What is the difference between an arithmetic and a geometric sequence?
In an arithmetic sequence, you add the same number each time (equal spacing). In a geometric sequence, you multiply by the same number each time (not equally spaced).
How are arithmetic sequences related to linear functions?
Each term in an arithmetic sequence corresponds to a point on a line when graphed. The common difference corresponds to the slope of that line.